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| | ! Name | | ! Name |
| | ! differential form | | ! differential form |
| − | ! differential form, E and B only
| |
| | ! integral form | | ! integral form |
| | |- | | |- |
| | | Coulomb's law of electrostatics, or [[Gauss's Law]]: | | | Coulomb's law of electrostatics, or [[Gauss's Law]]: |
| − | | <math>\nabla \cdot \mathbf{D} = \rho</math> | + | | <math>\nabla \cdot \mathbf{D} = \rho</math> |
| − | | <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon}</math>
| |
| | | <math>\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V</math> | | | <math>\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V</math> |
| | |- | | |- |
| | | Absence of magnetic monopoles: | | | Absence of magnetic monopoles: |
| − | | <math>\nabla \cdot \mathbf{B} = 0</math> | + | | <math>\nabla \cdot \mathbf{B} = 0</math> |
| − | | <math>\nabla \cdot \mathbf{B} = 0</math>
| |
| | | <math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math> | | | <math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math> |
| | |- | | |- |
| | | Faraday's Law of Induction: | | | Faraday's Law of Induction: |
| − | | <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math> | + | | <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math> |
| − | | <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
| |
| | | <math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math> | | | <math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math> |
| | |- | | |- |
| − | | Ampère's Law, or the Biot-Savart Law, | + | | Ampère's Law, or the Biot-Savart Law, plus displacement current: |
| − | :plus displacement current:
| |
| | | <math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math> | | | <math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math> |
| − | | <math>\nabla \times \mathbf{B} = \mu \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}</math>
| |
| | | <math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + | | | <math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + |
| | \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math> | | \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math> |
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| | | | |
| | ==Other Formulations== | | ==Other Formulations== |
| − | There are even more esoteric formulations (leading to more esoteric T-shirts) involving advanced concepts of differential geometry. | + | There are even more esoteric formulations (leading to more esoteric T-shirts) involving advanced concepts of differential geometry. See [http://www.answers.com/topic/differential-form]. |
| | | | |
| | == References == | | == References == |